Abstract. It is shown that every L2 -summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L 2 -summand vectors of a dual real Banach space can be determined by the L 2 -summand vectors of its predual; for every n ∈ ,ގ every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.2000 Mathematics Subject Classification. Primary 46B20, 46C05, 46B04.
Introduction and background.A vector e of a real Banach space X is said to be an L 2 -summand vector if there exists a closed vector subspace M of X such that X = ޒe ⊕ 2 M; in other words, λe + m 2 = λe 2 + m 2 for every λ ∈ ޒ and every m ∈ M. If e = 0, then the functional e * ∈ X * such that e * (e) = 1 and M = ker(e * ) is called the L 2 -summand functional associated to e. It satisfies e * =